What are some applications of subdirect product?

I have studied direct products. I know a few applications of direct products, like group isomorphism, etc. What are some applications of sub-direct product of groups?

• Exercise: a group is isomorphic to a (nontrivial) subdirect product of two groups iff it has two nontrivial normal subgroups with trivial intersection. (By nontrivial subdirect, I mean none of the two projections is injective on the subgroup.) – YCor Jan 24 at 3:46

Subdirect products arise naturally as intransitive subgroups of $$S_n$$, where they are subdirect products of the induced actions of the group on its orbits. Similarly for completely reducible linear groups.

• I was interested in the application in computer science. – I_wil_break_wall Jan 23 at 5:41
• Also, centralizers in the alternating group (e.g., of a 3-cycle) are usually naturally subdirect (fibre) products. – YCor Jan 24 at 3:47

A powerful tool in group theory is something called the fibre product. This is a special type of subdirect product of the group with itself.

Associated to a short exact sequence $$1\rightarrow N\rightarrow H\rightarrow Q\rightarrow 1$$, where $$\pi(H)=Q$$ is the natural map, is the fibre product $$P\subset H\times H$$, $$P:=\{(h_1, h_2)\mid \pi(h_1)=\pi(h_2)\}.$$

Clearly the diagonal component $$\Delta:=\{(h, h)\mid h\in H\}$$ is contained in $$P$$, so $$P$$ is a subdirect product.

The first application of fibre products which is know of is that there exist finitely generated subgroups of $$F_2\times F_2$$ which have insoluble membership problem: Firstly, let $$Q$$ be finitely presented and have insoluble word problem, then $$P$$ has insoluble membership problem. To see that $$P$$ is finitely generated, since $$Q$$ is ﬁnitely presented we have that $$N \subset H$$ is ﬁnitely generated as a normal subgroup, and then to obtain a ﬁnite generating set for $$P$$ one chooses a ﬁnite normal generating set for $$N\times\{1\}$$ and then appends a generating set for the diagonal $$\Delta\cong H=F_2$$.

A surprisingly powerful result, called the $$1$$-$$2$$-$$3$$ theorem, gives conditions on the fibre products to be finitely presentable, see: G. Baumslag, M.R. Bridson, C.F. Miller III, H. Short, Fibre products, non-positive curvature, and decision problems, Comm. Math. Helv. 75 (2000), 457–477.

A super-powerful application of fibre products is the following (the application is the main result of a paper in the Annals of Mathematics, one of the top journals - undisputed top 4 journal, disputed no. 1). If $$\Gamma$$ is a group then $$\widehat{\Gamma}$$ denotes its profinite completion.

Question (Grothendieck, 1970). Let $$\Gamma_1$$ and $$\Gamma_2$$ be ﬁnitely presented, residually ﬁnite groups and let $$u :\Gamma_1\rightarrow \Gamma_2$$ be a homomorphism such that $$\widehat{u} :\widehat{\Gamma}_1\rightarrow \widehat{\Gamma}_2$$ is an isomorphism of proﬁnite groups. Does it follow that $$u$$ is an isomorphism from $$\Gamma_1$$ onto $$\Gamma_2$$?

Theorem (Bridson, Grunewald, 2004, Ann. Math., link). There exists a short exact sequence $$1\rightarrow N\rightarrow\Gamma\rightarrow Q\rightarrow 1$$ where $$\Gamma$$ has a whole host of nice properties, and where $$P$$ and $$\Gamma$$ are a counter-example to Grothendieck's question: $$P$$ and $$\Gamma$$ are finitely presentable, with $$\widehat{P}\cong \widehat{\Gamma}$$ but $$P\not\cong \Gamma$$.

• @Shaun regarding your numerals edit, you might be interested in this tex.SE question. – user1729 Jan 22 at 19:10
• Thank you, @user1729; that'll save me some time in future! It's nice to know a useful convention :) – Shaun Jan 22 at 19:45
• "A powerful tool in group theory, developed over the last 20 or so years but has a longer history". This sounds weird. I'd certainly not say that the tool of fibre products was developed in this period. Also if I prove a result about direct products I'm not developing the tool of direct product. – YCor Jan 24 at 3:43
• @YCor I'll remove the statement with the dates. However, I can only think of three results pre-Baumslag, Bridson, Miller, Short which use fibre products (but I think they invented the name?). So do you think it would be correct to say that they "came to the fore" in the past 20 years? – user1729 Jan 24 at 9:41

Typically, when you know that an object is a subdirect product of other objects, you know that it inherits all properties that are passed from products of those objects to their subsets. For example, a subdirect product of abelian groups must be abelian (and the same for any other group identity).

• This seems to just be a property of a "subgroup of the direct product", which is a weaker property than being a subdirect product. – Ted Jan 22 at 17:07